Nous introduisons la catégorie des espaces boréliens-topologiques, appelée BT-catégorie : c'est un cadre naturel pour la classification mesurable des feuilletages et des laminations usuels. Nous nous intéressons au cas des laminations de dimension deux. Nous prouvons les deux résultats principaux suivants :

1. Une lamination borélienne par plans est la suspension d'une action de $\mathbb{Z}^2$ sur un espace de Borel standard si et seulement si elle est hyperfinie.

2. Toute lamination par surfaces admettant une structure complexe parabolique mesurable est moyennable.

1. Une lamination par plans, cylindres et tores est $\mu$-moyennable si et seulement si elle admet une métrique de Riemann plate le long de $\mu$-presque chaque feuille.

It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for any genus-2 translation surface which is not a Veech surface there are uncountably many minimal but not uniquely ergodic directions. The theorem can be applied to certain billiard tables.

We consider properties of Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. This is a slight generalization of the situation first considered by Fornaess and Sibony in Random iterations of rational functions. Ergod. Th. & Dynam. Sys.11 (1991), 687–708. The classical definition of hyperbolicity has a natural extension to our setting, and we show that many of the classical results concerning hyperbolic Julia sets can indeed be generalized. We introduce two important ideas in order to prove these results—the use of Cantor diagonalization, and how the Julia set moves in the Hausdorff topology as the sequence generating it changes.

In this article we study the asymptotic dynamics of highly oscillatory solutions for the unbalanced Allen–Cahn equation with a slowly varying coefficient. We describe the underlying structure of these solutions through a function we call the adiabatic profile, which accounts for the asymptotic area covered by the solutions in the phase space. In finite intervals, we construct solutions given any adiabatic profile. In the case of a periodic coefficient we show that the system has chaotic behavior by constructing high-frequency complex solutions which can be characterized by a bi-infinite sequence of real numbers in $[c_1,c_2]\cup\{ 0\}\ (0<c_1<c_2)$.

We prove quantitative equidistribution results for nilflows on compact three-dimensional homogeneous nilmanifolds by a method based on renormalization and invariant distributions for nilflows. As an application we obtain a dynamical proof of quantitative equidistribution results for the sequence P(n) (mod 1), where $P(X)\in \mathbb{R}[X]$ is a quadratic polynomial with generic leading coefficient. Bounds on theta sums, that is, Birkhoff sums of the exponential function along such sequences, were proved by Hardy and Littlewood and in optimal form by Fiedler, Jurkat and Körner.

Theorem 1.1 in [4] is false: the hypothesis $\sum v_n(g)<\infty$ needs to be replaced by a stronger condition. A corrected version of Theorem 1.1 is given below; a counterexample to the original result is given in [5].

An example of non-unique g-measures is constructed. The function g satisfies a certain summability condition, thereby providing a counterexample to the claim in an earlier paper that all such g have a unique g-measure.

We carry the argument used in the proof of Denjoy's theorem over to the quasiperiodically forced case. Thus we derive that, if a system of quasiperiodically forced circle diffeomorphisms with bounded variation of the derivative has no invariant graphs with a certain kind of topological regularity, then the system is topologically transitive.

We will show that an equivalence relation on a Cantor set arising from a two-dimensional substitution tiling by polygons is affable in the sense of Giordano, Putnam and Skau.

In this paper we study the existence of analytic families of reducible linear quasi-periodic differential equations in matrix Lie algebras. Under suitable conditions we show, by means of a Kolmogorov–Arnold–Moser (KAM) scheme, that a real analytic quasi-periodic system close to a constant matrix can be modified by the addition of a time-free matrix that makes it reducible to constant coefficients. If the system depends analytically on external parameters, then this modifying term is also analytic.

As a major application, we prove the analyticity of resonance tongue boundaries in Hill's equation with a small quasi-periodic forcing. Several consequences for the spectrum of Schrödinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schrödinger operators with a small real analytic and quasi-periodic potential has all spectral gaps open and, therefore, it is a Cantor set. Some other applications are included for linear quasi-periodic systems on $so(3,\mathbb{R})$ and $sp(n,\mathbb{R})$.

Let f be a transcendental meromorphic function with finitely many poles such that the finite singularities of f-1 lie in a bounded set. We show that the Julia set of f has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions.

Let S be a compact surface without boundary, with Euler characteristic $\chi(S)>0$, and let $K \subset S$ be a compact set such that each component of K is cellular. The aim of this paper is to prove that, if $f:S \setminus K \rightarrow S \setminus K$ is a homeomorphism which has a certain instability property in the proximity of K, then the set of periodic orbits of f is non-empty. This result give us some corollaries:

1. there is no minimal homeomorphism $f:S^2 \setminus K \rightarrow S^2 \setminus K$ for K an arbitrary compact set;

2. if $f:S^2 \rightarrow S^2$ is an area-preserving homeomorphism, then $K=\text{cl(Per($f$))}$ is not an isolated invariant set.

We study the algebraic properties of automorphism groups of two-sided, transitive, countable-state Markov shifts together with the dynamics of those groups on the shift space itself as well as on periodic orbits and the 1-point-compactification of the shift space. We present a complete solution to the cardinality question of the automorphism group for locally compact and non-locally compact, countable-state Markov shifts, shed some light on its huge subgroup structure and prove the analogue of Ryan's theorem about the center of the automorphism group in the non-compact setting. Moreover, we characterize the 1-point-compactification of locally compact, countable-state Markov shifts, whose automorphism groups are countable and show that these compact dynamical systems are conjugate to synchronized systems on doubly transitive points. Finally, we prove the existence of a class of locally compact, countable-state Markov shifts whose automorphism groups split into a direct sum of two groups, one being the infinite cyclic group generated by the shift map, the other being a countably infinite, centerless group, which contains all automorphisms that act on the orbit-complement of certain finite sets of symbols like the identity.

We propose to study the renormalization operator acting on critical $\mathcal{C}^r$ circle mappings. (More precisely, the operator acts on critical commuting pairs.) Assuming that there is a Banach manifold of critical analytic commuting pairs on which the renormalization operator acts hyperbolically (with non-trivial hyperbolic attractor), we prove that, for r > 2, the operator remains hyperbolic with the same expanding subspaces when acting on $\mathcal{C}^r$ commuting pairs. By this we mean that the tangent renormalization operator admits a hyperbolic splitting with the same unstable subbundle.